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Friday, July 27, 2012

So when Dave and I filmed the first episode, we actually filmed two. Given where the whole thing has gone, more episodes seem unlikely from us. (Unless we could do something meta with it...)

Of course, this also means whatever you disliked about the first one you will also dislike at least as much in this one. Audio quality, snark, the guy on the left...

Editing this it struck me that really Sal Khan is a tutor. He's like the strong student in the class that is willing to share with other students how he sees it. So his mathematical viewpoint is a bit procedural, and his understanding of the ideas comes across as a novice. A skilled novice, but without depth. It was his discussion of matrices that really sealed this for me. Undoubtedly there are a lot of teachers who see a matrix as just a table of numbers, but there is so much more. It may be a teaching decision to not share that 'much more' in an introduction, or it may be he's just sharing his viewpoint.

If this is your first exposure to MTT2K, there's still time to look away. If you don't, you might want to read Dave's two posts on how it came to be (one and two) or watch the first episode. I have a (first crack at a) storify with some of the brouhaha surrounding the first.

I'll just say here, I'm not against Khan Aademy, I'm not jealous, and I'm not against flipped classrooms. I am for quality materials, intentional teaching and learning, and open discussion of ideas. I do believe satire is an appropriate response to exaggerated publicity and overhype. If I could sing, I'd explore a Tom Lehrer style song on the matter. It's gratifying that the first episode started a big discussion, but at some level this is two guys goofing around to make a point about good use of resources.

In contrast to many places discussing Khan Academy, the comments are open. I'll ask you to be as civil as you can, though, please. Snark, satire and sarcasm are strictly permitted.

Friday, July 20, 2012

I got to give a whiz-bang 60 minute (with an option for 30 extra minutes) intro to GeoGebra at the New Tech network conference this week. 50 plus tech-savvy teachers... so it was good. I am always worried that people expect me to tell them about GeoGebra for an hour, when purpose is to get them started using it on the spot, in ways that make sense of their potential use. (Note that if you are in driving distance, I am more than happy to come do this at your school. No GeoGebra lectures, however.)

Purposes. So what are the ways that people make use of it? Oh, let me count them:

World's best graphing calculator. (A little weak on statistics and CAS, but that's improving quickly.) For you and your students. For algebra, calculus, or geometry.

Demonstration tool. Project a great visualization on your screen to show to or discuss with students.

Focused mathematical activity for students.

Open-ended inquiry tool. Pose a question and let students investigate.

Requirements. The AMAZING thing about this tool is that with version 4.0, all of these are accessible to teachers in that 60-90 minute start up.

Open the program, start typing equations on the input bar.

Needs some quick familiarity with the tool bar to make your image, then File > Export > Graphics View As A Picture.

GeoGebraTube. If you have not looked at this, you are missing out. 14,000 sketches and counting; free accounts, search, likes, tagging and you can collect them in teacher mode or show collections in student friendly mode. This is why you need minimal expertise to start using the program deeply. If you can run YouTube and you are a teacher, you can do this.

See #3.

Students today are geared for this kind of tool. You give them access, they'll figure things out about it that I don't know.

Really, any training beyond that first 90 min. is about if you want to become proficient in number 5, or if you want to be designing your own activities. Some teachers are doing that anyway by the end of an hour, most by the end of a half day. Once you start using it, there's a big danger of being sucked in by the possibilities of what you can make. The power of dynamic examples is as much greater than static electronic images as static electronic images were than hand drawn. (My opinion. No research. Actually yes research, but they would never quantify so crazily.)

After my session, I got to go to Geoff Krall's (@emergentmath) session on formative assessment. He was using the MARS MAP (Mathematics Assessment Resource Service - Mathematics Assessment Project) materials. In particular, he used the Ferris Wheel lesson to get us collaborating and specific in discussion.

As we discussed, it really got me thinking about how I would use the task early on. It would make a good project or assessment, I think, but what about as an inquiry? The basic problem was to make a symbolic model [find a, b and c for a+b*cos(ct)] for the height of a car on a specific Ferris wheel. Then there was a card sort which got students comparing context, equation and graphs.

I've given many explorations before that got students experimenting with parameters to see the effect on graphs, but I love the idea of tying it to a context. That doubles up on the intuition they can apply - physical and visual. If the students had access to that, they might be able to do enough trials to start to generalize. Even without much trigonometry understanding, it's a nice context for graph transformations. For me, these kind of thoughts now lead to GeoGebra. I made a quick sketch, with the Ferris wheel in a 2nd graphics window, and was delighted to find that even the 2nd window worked on GeoGebraTube.

But since then I thought it would be worthwhile to develop a bit more. Both to familiarize myself with using the 2nd graphics window and to make the single model into a reusable activity. I knew I wanted to have either a customizable or random Ferris wheel, some animation of the situation and a way for the students to enter the equation.

That bore some thought: sliders, input boxes for parameters or an input box for the function? Sliders are best for seeing continuously linked examples, but can make a problem like this too easy! The input boxes for the parameters helped support the idea of structure, require some thinking before making a new guess, and don't require as much typing as entering the whole function. Plus you can isolate one parameter and just adjust that. That might be a positive or negative. It feels like a support for learners early in this, by encouraging them to focus on one parameter at a time.

The trick to working on two graphics views is the advanced tab of object properties. You can use any tools from the main window. Just select the tool and then use it in the 2nd graphics window. The objects you make there show up in the algebra view. But when you edit things, or create them in the input bar, they migrate or appear in the first graphics window. The solution is in the object properties, advanced tab; just check the box you need. Note that you can have something appear in both ... there just has to be a cool use of that.

I don't think there's anything else too tricky about the sketch. I used the Function[ , , ] to get the modeled equation to move with the tracing point, the ZoomIn[1] command on the button to clear traces, and the UpdateConstruction[] command to reset the Ferris wheel dimensions. (I had slick graphics window dimensions based on the Ferris wheel, but then the ZoomIn[1] command doesn't work. Ultimately I thought it was better to see the Ferris wheel changing sizes anyway.)

Unable to display content. Adobe Flash is required.

The sketch is on GeoGebraTube: Teacher page for download or Student worksheet for in browser use. You have to click in the main window to get the animation button to show.

Thursday, July 12, 2012

I had one of those great twitter moments yesterday, completely by generosity of tweeps. This captures one aspect of why I introduce twitter to our preservice teachers in hopes that they will either enter in or give it a go later.

My summer intermediate algebra class is leisurely (12 weeks instead of 6) picking its way through the summer. Linear, quadratics, exponentials with tons of technology use, simulation and experiment and a dash of art. And then comes logarithms.

My emphasis on introduction is as a way to undo exponentiation. Not a big emphasis on inverse functions, because though we're using the language, we haven't really dove into function language yet. But doing and undoing we talk about a lot.

But after that reasonably good start, we come up against the log rules. Our introduction was Kate Nowak's log law introduction (which I found through Sam Shah's Virtual Filing Cabinet; why bookmark? Let Sam do it for you.) The idea is that you see lots of examples and then try to generalize into a pattern, which is the log law. Nice pedagogy!

Yesterdays lesson included connecting the exponent rules to the log laws.

Tough going. In terms of gradual release, we had to back up to a lot of teacher support. But the most useful law was the toughest. So I came back from class and just tweeted, to ... vent, I guess. Commiserate. (Which is definitely a purpose of twitter.)

Wasn't expecting any real response. But then, the great feedback and constructive suggestions began immediately.

This is where we should start. We - as a teaching culture - are so steeped in general then specific, abstract before concrete, that this is a good first check.

They did understand the multiplication to addition rule better. Look for areas where students understand and build off of those. Connections strengthen learning.

This is the connection I'm going to follow up on. Look for a way to get it across.

Think about the broader context. How the logarithm lives as an inverse function, with a nice concrete place to start.

Support that this is challenging, and not me just being stupid. Of course, I'm happy when people point out I'm just being stupid, too, since I don't want to be stupid.

Neat post. It gives a lot of good thinking about introducing logs, being intentional, and paying attention to students making sense of notation. This is very close to how I introduce logs, and how it looks like Kate Nowak introduces them, too. I use to think this was inappropriate for high school but okay for college (because of their bad early experiences), but I'm beginning to think that's how we should handle bad math notation. Use constructive notation and then transition the students to traditional.

I don't think this is accessible (YET), but I'm going to mention this, too. Part of the class is getting the students more comfortable with symbolic methods, and I like how this emphasizes the operational part of logarithms.

A reminder of where to start. This is the fundamental relationship for logarithms, and connecting back to it well and often is important for learning in both directions.

So now I have a place to go Monday, following this up as I promised the students. Maybe I would have had some of these insights on my own - but I don't have to work and think alone about my teaching or the mathematics. It also really sparked some thinking to me about whether logs should be introduced as a function or an operation. I feel like exponentiation goes from being notation, to an operation, to a function. Logarithms could do the same. I'm thinking:

So then 2v2^3=3. As it should be. I'm kind of joking?

Anyway, excellent teaching advice. I am so glad to be connected to these excellent teachers. And even though I can't go to Twitter Math Camp, I still get to have the home game to play. Why don't you play along?

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